MHB Does the Intermediate Value Theorem Apply If h(a) and h(b) Have Opposite Signs?

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The Intermediate Value Theorem (IVT) applies when a continuous function h(x) has opposite signs at two points, h(a) and h(b). In this case, with h(b) ≤ 0 and 0 ≤ h(a), the conditions of the IVT are satisfied. This means there exists at least one point c in the interval (a, b) where h(c) = 0. The continuity of h(x) ensures that all values between h(b) and h(a) are achieved. Thus, the conclusion is valid that there is a c in (a, b) such that h(c) = 0.
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I have a continuous function h(x) and the inequality h(b)<=0<=h(a). Can I apply IVT?
 
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Well, what are the hypotheses of the IVT? Are they satisfied in your case?
 
Ackbach said:
Well, what are the hypotheses of the IVT? Are they satisfied in your case?
Isn't it just that h(x) is continuous and if u is a number between h(b) and h(a),
h(b) < u < h(a), then etc etc. I didn't think I could, but I just wanted to see a variation of it.
 
TheDougheyMan said:
Isn't it just that h(x) is continuous and if u is a number between h(b) and h(a),
h(b) < u < h(a), then etc etc. I didn't think I could, but I just wanted to see a variation of it.

Exactly right. So you can conclude that there is a $c \in (a,b)$ (I'm assuming $a<b$) such that $h(c)=0$.
 

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